Several studies have compared the in-distribution (ID) and out-of-distribution (OOD) performance of models in computer vision and NLP. They report a frequent positive correlation and some surprisingly never even observe an inverse correlation indicative of a necessary trade-off. The possibility of inverse patterns is important to determine whether ID performance can serve as a proxy for OOD generalization capabilities.This paper shows that inverse correlations between ID and OOD performance do happen with multiple real-world datasets, not only in artificial worst-case settings. We explain theoretically how these cases arise and how past studies missed them because of improper methodologies that examined a biased selection of models.Our observations lead to recommendations that contradict those found in much of the current literature.- High OOD performance sometimes requires trading off ID performance.- Focusing on ID performance alone may not lead to optimal OOD performance. It may produce diminishing (eventually negative) returns in OOD performance.- In these cases, studies on OOD generalization that use ID performance for model selection (a common recommended practice) will necessarily miss the best-performing models, making these studies blind to a whole range of phenomena.